Nel's category theory based differential and integral Calculus, or did Newton know category

a r X i v :m a t h /0504565v 1 [m a t h .G M ] 28 A p r 2005Nel’s category theory based

differential and integral Calculus,or Did Newton know category theory ?Elem´e r E Rosinger Department of Mathematics and Applied Mathematics University of Pretoria Pretoria 0002South Africa eerosinger@ Abstract In a series of publications in the early 1990s,L D Nel set up a study of non-normable topological vector spaces based on methods in category theory.One of the important results showed that the classical oper-ations of derivative and integral in Calculus can in fact be obtained by a rather simple construction in categories.Here we present this result in a concise form.It is important to note that the respective differentiation does not lead to any so called generalized derivatives,for instance,in the sense of distributions,hyperfunctions,etc.,but it simply corresponds to the classical one in Calculus.

Based on that categorial construction,Nel set up an infinite dimen-sional calculus which can be applied to functions defined on non-convex domains with empty interior,a situation of great importance in the solution of partial differential equations.

1.The Setup

The presentation follows mainly Nel [1],where further details as well